Question: Solve for $x$ and $y$ using elimination. $\begin{align*}8x-9y &= -4 \\ 4x+3y &= -8\end{align*}$
We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $2$ $\begin{align*}-8x+9y &= 4\\ 8x+6y &= -16\end{align*}$ Add the top and bottom equations. $15y = -12$ Divide both sides by $15$ and reduce as necessary. $y = -\dfrac{4}{5}$ Substitute $-\dfrac{4}{5}$ for $y$ in the top equation. $8x-9( -\dfrac{4}{5}) = -4$ $8x+\dfrac{36}{5} = -4$ $8x = -\dfrac{56}{5}$ $x = -\dfrac{7}{5}$ The solution is $\enspace x = -\dfrac{7}{5}, \enspace y = -\dfrac{4}{5}$.